Basics
- If a plane intersects a cone, it makes one of the following
- 0-1 points
- 0-2 straight lines
- Circle
- Parabola
- Ellipse
- Hyperbola
- Standard equation for all the things that this intersection makes
- $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$
- Equations to change angle of cartesian plane, assuming B is not 0
- $$ x = x\prime\cos(\alpha) - y\prime\sin(\alpha) $$
- $$ y = x\prime\sin(\alpha) + y\prime\cos(\alpha) $$
- $$ \tan(2\alpha) = \frac{B}{A-C} $$
- Eccentricity (e)
- the distance to the focus (F) divided by the distance to the directrix (DD')
Circle
- $$ (x - h)^2 + (y - k)^2 = r^2 $$
- $$ e = 0 $$
- $$ a = b $$
-
Parabola
- distance to DD' = distance to f
- $$ e = 1 $$
- $$ LR = 4p $$
- $$ 4p(y - k) = (x - h)^2 $$
-
- It is up (+) or down (-)
- $$ 4p(x - h) = (y - k)^2 $$
-
- It is left (-) or right (+)
Ellipse (Paraboloid)
- sum of distance to foci = 2a (major axis)
- minor axis = 2b
- $$ c = \pm\sqrt{a^2 - b^2} $$
- $$ e = \frac{distance(f)}{distance(DD\prime)} $$
- $$ 0 < e < 1 $$
- $$ a > b $$
- $$ e = \frac{c}{a} $$
- $$ LR = \frac{2b^2}{a} $$
- $$ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 $$
-
- It is left-right
- $$ a = 2, b = 1, h = k = 0 $$
- $$ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 $$
-
- It is up-down
- $$ a = 2, b = 1, h = k = 0 $$
Hyperbola
- Triangle (?) distance from point on the curve to f, f' = 2a
- conjugate ax (?), TA = 2b
- Transverse m (?), CA = 2a
- Transverse ax = 2a ?
- $$ c = \pm\sqrt{a^2 + b^2} $$
- $$ e > 1 $$
- $$ e = \frac{c}{a} $$
- $$ LR = \frac{2b^2}{a} $$
- $$ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 $$
-
- It is left-right
- $$ a = b = h = k = 0 $$
- $$ -\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 $$
-
- It is up-down
- $$ a = b = h = k = 0 $$
- Something cool to solidify concept
- I can't graph these equations as they are as they are implicitly defined (as in, it isn't y = func(x))
- To solve this conundrum, I set $$ y^2 = ... x^2 ...$$
- then: $$ y = \sqrt{...x^2...} $$
- $$ y = -\sqrt{...x^2...} $$
- Which should seem like the familiar $$ y = \pm func(x) $$
- And graphed each y equation